Positive invariant set

In mathematical analysis, a positively invariant set is a set with the following properties:

Given a dynamical system $\dot{x}=f(x)$ and trajectory $x(t,x_0) \,$ where $x_0 \,$ is the initial point. Let $\mathcal{O} \triangleq \left \lbrace x \in \mathbb{R}^n| \phi (x) = 0 \right \rbrace$ where $\phi$ is a real valued function. The set $\mathcal{O}$ is said to be positively invariant if $x_0 \in \mathcal{O}$ implies that $x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0$

Intuitively, this means that once a trajectory of the system enters $\mathcal{O}$ , it will never leave it again.

References

Dr. Francesco Borrelli